Hyperbolic Functions: Properties & Applications

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  • 0:02 Hyperbolic Functions…
  • 0:49 Identifying Hyperbolic…
  • 2:41 Examples of Hyperbolic…
  • 4:47 Other Uses of…
  • 5:30 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Although related to trigonometric functions, hyperbolic functions have special properties. In this lesson, you'll explore the properties of hyperbolic functions and their usage in both theoretical and real-world applications.

Hyperbolic Functions in Real Life

You and your friend had planned to walk to the movies together, but now it's five minutes to show time and no sign of him. Suddenly there he is, running on the sidewalk in front of your house, and it looks like stopping for you is not on his agenda. You jump off your porch and start to run. Heading towards your friend, you constantly change direction until you catch up.

Looking back at your footsteps, your friend remarks, 'Hey, you traced out an interesting curve.' To which you respond, 'For sure. It's based on hyperbolic functions.' Okay, okay, you probably wouldn't have said this. That is, not until you've completed this lesson.

Hyperbolic functions occur in all sorts of places. Let's talk about their properties and some of their cool applications.

Identifying Hyperbolic Functions

Hyperbolic functions are defined in terms of exponentials, and the definitions lead to properties such as differentiation of hyperbolic functions and their expansion as infinite series. They're written like the trig functions cosine (cos), sine (sin), tangent (tan), but they have an 'h' at the end. Cosh is pronounced 'kosh;' sinh is pronounced 'sinch;' and tanh is usually read as 'tan h' but sometimes we say 'tanch.'

Let's look at the graphs of cosh(x) and sinh(x):


That curve you traced is called the pursuit curve or the tractrix curve. The equations relating the x and y positions as a function of time both involve hyperbolic functions. In the following graph, do you see the square (you) pursuing the circle (your friend)?


Note the hyperbolic functions cosh(x) and tanh(x).

Identifying Hyperbolic Functions

The cosh(x) and sinh(x) functions may be defined in terms of exponential functions. Note that x is the independent variable.



Similarities exist between trig functions and hyperbolic functions. For example, tangent is sine divided by cosine. For hyperbolic tangent,


Trig functions have special names for their reciprocals. Secant, abbreviated sec, is one over cosine; cosecant, abbreviated csc, is one over sine; and cotangent, abbreviated cot, is one over tangent. The hyperbolic versions of these functions have an 'h' added to the end. Thus, we have sech(x), csch(x), and coth(x), as you can see:




Here are some other hyperbolic function properties.


In terms of their derivatives,



As a series,



Examples of Hyperbolic Functions

Let's use hyperbolic function properties.

Example 1

When a flexible cable is suspended between two towers, the relevant equation is:


We would like to show you that y = cosh(x) is a solution to this equation.

To solve this problem we'll start by assuming that y = cosh(x) is the solution.

Recall that the first derivative is called y prime and the second derivative is called y double-prime.

Starting with our assumption that y is cosh(x), y prime is the derivative of cosh(x). From the properties, the derivative of cosh(x) is sinh(x). Thus, y prime is sinh(x).

Continuing, y double-prime is the derivative of y prime. That means that y double-prime is the derivative of sinh(x). From the properties, we know that the derivative of sinh(x) is cosh(x). Thus, y double-prime is cosh(x).

You can see what we have so far:


Now, let's substitute into our equation. We get the following:


If we square both sides, we then arrive at the following:


Now, through some simple algebra we get this equation


which is one of our properties.

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